Constructing ghost-free degenerate theories with higher derivatives

Transcription

1 Constructing ghost-free degenerate theories with higher derivatives Hayato Motohashi Center for Gravitational Physics Yukawa Institute for Theoretical Physics HM, Suyama, PRD 91 (2015) 8, , [arxiv: ] HM, Noui, Suyama, Yamaguchi, Langlois, JCAP 1607 (2016) 07, 033, [arxiv: ] HM, Suyama, Yamaguchi, [arxiv: ]; in preparation YITP long-term workshop Gravity and Cosmology 2018

2 Scalar-tensor theories Ostrogradsky Ghost Healthy theories with arbitrary higher-order derivatives Dark energy Inflation GLPV Healthy theories with 2nd-order derivatives DHOST / EST Horndeski theory DGP Brans-Dicke Extended Galileon! ", $ % & '( ) ' ) ( " +(", $)!(%)

3 Ostrogradsky theorem for!(# %, # %, # % )!!(# %, # %, # % ) # % # % * and + 1,. Woodard, / % (kinetic matrix) üostrogradsky theorem det / 0? is unbounded

4 üostrogradsky theorem det is unbounded Hamiltonian analysis! "#! % ', % ', ) ' + + ' (% ' ) ' ) ) H ) H ) ' 8 9, : 9, ; 9 < 9, 9, > 9 3 '? AB C AD F AD G

5 üostrogradsky theorem det ; 0 > is unbounded Hamiltonian analysis! "#! % ', % ', ) ' + + ' (% ' ) ' ) Canonical momenta ) ' A B, C B, + ' / ' ' + ' 6 ' 0 0? det 3 0 % ' % ' (/, %, )) Primary constraints (C1) D B, E B, 6 ' ; 'F 0H 1 0I 3

6 üostrogradsky theorem det? 0 A is unbounded Hamiltonian analysis! "#! % ', % ', ) ' + + ' (% ' ) ' ) Canonical momenta ) ' H I, J I, + ' / ' ' + ' 6 ' 0 5 ' + + ', ' 8 Second class. No secondary constraints (C2) ; healthy + ; ghost DOFs 0B det 3 02 C 0 % ' % ' (/, %, )) Primary constraints (C1) K I, L I, 6 '? '8 0E 1 0F 3 0F C

7 üostrogradsky theorem det 0 8 is unbounded Hamiltonian analysis! "#! % ', % ', ) ' + + ' (% ' ) ' ) Canonical momenta ) ' G H, I H, + ' / ' ' + ' 6 ' 0 Hamiltonian 0@ det 3 02 A 0 % ' % ' (/, %, )) Primary constraints (C1) /, %, ) + 5 ' % ' 5 ' shows up only linearly. 8 is unbounded. J H, K H, 6 ' 'B 0D 1 0E 3 0E A

8 üostrogradsky theorem det! 0 ' is unbounded? No-ghost condition?! "# 0 ' is bounded

9 üostrogradsky theorem det! 0 ' is unbounded ü! "# 0 ' is bounded? No-ghost condition? Different! "# 0 ' is bounded... though it is a part of no-ghost conditions 1st degeneracy condition (DC1)

10 !(# %, # %, # % ) ) ) %. % HM, Suyama, DC1: %> 0 Additional C1: Ψ % 3 % 5 % (., #) 0 ü Fixed Still - % is not fixed. # % 7 8, 9 8, : % 3 %, ; 8, < %

11 !(# %, # %, # % ) ) ) %. % HM, Suyama, ; <, <, > % %A 0 3 %, - %,? % Additional C1: Ψ % 3 % 5 % (., #) 0 ü Fixed DC2: B %A {Ψ %, Ψ A } 0 C2: Υ 8 - % 9 % (., #) 0 ü Fixed ü We eliminated all the ghosts. ) is bounded. # % ü The most general Lagrangian:! 9 # %, # %

12 üostrogradsky theorem det! 0 ( is unbounded üno-ghost condition (DC1 & DC2)! "# 0 & & "# 0 ( is bounded! "# /0 1 /2 4 /2 5 & "# /0 1 /2 4 /2 5 /0 1 /2 5 /2 4

13 üostrogradsky theorem updated det! 0 or det & 0 ( is unbounded üno-ghost condition (DC1 & DC2)! "# 0 & & "# 0 ( is bounded! "# /0 1 /2 4 /2 5 & "# /0 1 /2 4 /2 5 /0 1 /2 5 /2 4

14 üostrogradsky theorem updated det! 0 or det ( 0 7 is unbounded üno-ghost condition (DC1 & DC2)! "# 0 & ( "# 0 7 is bounded Highest Next-highest ü EL eq! $%# "# +! "# + ( $)# " "# terms up to $ 2nd-order system

15 Arbitrary higher-order derivatives!!($ % &, $ % &(),, $ % ) $ % $ %, and - 1, 0 HM, Suyama, % : 9, ; % : 9< 47 : < üostrogradsky theorem updated det 1 0 or det ; 0 F is unbounded 1 %2 0 ; %2 0 ü $ %(I&) from EL eq ü $ %(I&()) from EL eq Highest Next-highest Still remain ghosts from lower (> 2) derivatives.

16 Eliminating Ostrogradsky ghost ü!(#, #, #)!(#, #) ü!(# ), # ), # ) )!(# ), # ) )!(# ) *, # ) *+,,, # ) ) HM, Suyama,

17 Eliminating Ostrogradsky ghost ü!(#, #, #)!(#, #) ü!(# ), # ), # ) )!(# ), # ) )!(# ) *, # ) *+,,, # ) ) HM, Suyama, !(#, #, #; /, /) HM, Noui, Suyama, Yamaguchi, Langlois, !(#, #, #; / 0, / 0 ) # !(# ), # ), # ) ; / 0, / 0 ) # )

18 HM, Noui, Suyama, Yamaguchi, Langlois, !(# %, # %, # % ; ) *, ) * ) Hamiltonian analysis!,-! / %, / %, # % ; ) *, ) * + 1 % (/ % # % ) Canonical momenta # % A B, 1 % C?, D?, E B, : % 3 %! *! % 1 % : % 0 Hamiltonian Primary constraints (C1) < < (3, /, #, 6, )) + 8 % / % 8 % shows up only linearly. < is unbounded.

19 HM, Noui, Suyama, Yamaguchi, Langlois, !(# %, # %, # % ; ) *, ) * ),,. + 0 % 1 % DC1:! 6 %, 0 %, D B, E % J K J L! M N J K! PQ M N M O! M O J L 0 Additional C1: Ψ % 6 % 8 % (1, #, 9, )) 0 DC2: det! M N M O 0 ü Fixed F %G {Ψ %, Ψ G } 0 C2: Υ < 0 % % (1, #, 9, )) 0 ü Fixed ü We eliminated all the ghosts., is bounded. ü EL eqs 2nd-order system # % A B, C % ü Applies for a wide class of theories

20 Applications ü Field theory in flat spacetime ü SU(2) ü Boson-Fermion ü Scalar-tensor theories ü Vector-tensor theories Crisostomi, Klein, Roest, Allys, Peter, Rodriguez, Kimura, Sakakihara, Yamaguchi, Langlois, Noui, , Crisostomi, Koyama, Tasinato, Achour, Langlois, Noui, Achour, Crisostomi, Koyama, Langlois, Noui, Tasinato, Kimura, Naruko, Yoshida, ü Tensor theories Crisostomi, Noui, Charmousis, Langlois,

21 Eliminating Ostrogradsky ghost ü!(#, #, #)!(#, #) ü!(# ), # ), # ) )!(# ), # ) )!(# ) *, # ) *+,,, # ) ) HM, Suyama, ü!(#, #, #; /, /) HM, Noui, Suyama, Yamaguchi, Langlois, ü!(#, #, #; / 0, / 0 ) # ü!(# ), # ), # ) ; / 0, / 0 ) # )

22 Eliminating Ostrogradsky ghost ü!(#, #, #)!(#, #) ü!(# ), # ), # ) )!(# ), # ) )!(# ) *, # ) *+,,, # ) ) HM, Suyama, ü!(#, #, #; /, /) ü!(#, #, #; / 0, / 0 ) # ü!(# ), # ), # ) ; / 0, / 0 ) # ) !(56, 5, 5, 5; / 0, / 0 ) HM, Noui, Suyama, Yamaguchi, Langlois, HM, Suyama, Yamaguchi, ; in prep.!(567, 5 7, 5 7, 5 7 ; # ), # ), # ) ; / 0, / 0 )!(# 0 8 *9,, ; # 0 8:; * 0, ; ; # <, # 0 <)

23 HM, Suyama, Yamaguchi, Quadratic model with!"#, % ' ( * +, #-!"#!"- + * + / #-! #! - + * + 1 #-! #! - + * + 2 #-! #! #-!"#! #-! #! - + * + 5 '6% ' % 6 + * + 7 '6% ' % '6 % ' % #'!"# % ' Equivalent form ( :; ( <, <,,!, %, % + > #! # # # ( # < # )!"#! # #! Canonical momenta C D E, #- < #' % ' + 3 #- < - F ' 9 #' < # + 5 '6 % '6 % 6 C G #, I J E > # K LE 0, K NE 0 det 5 '6 0 det 1 #- 0! # S % ' (S) Primary constraints (C1)

24 HM, Suyama, Yamaguchi, Quadratic model with!"#, % ' ( ( * +, -./ # # We eliminated all the ghosts?! #! # >,? >, A B, C #, D # DC1: J #K L #' M 'N L NK 0, 7.,, -., 0 1., E B, F G., F H. Additional C1: Ψ #, 7. 0 ü Fixed DC2: {Ψ #, Ψ K } 2 R #K 0 C2: Υ #, -. 0 ü Fixed DC3: {Υ #, Ψ K } S #K 0 C3: Λ # ü Fixed No!

25 HM, Suyama, Yamaguchi, Quadratic model with!"#, % ' ( ( * +, -./ # # All DCs and Cs! #! # / #, 4 5, 6 5, 7 8, 9 #, : #, ;.,, -., 0 1., < 8, >.,?. (: linear in / # Hidden ghosts appeared (

26 HM, Suyama, Yamaguchi, Quadratic model with!"#, % ' ( ( * +, -./ # # All DCs and Cs! #! # / #, C D, E D, F G, H #, I #, J.,, -., 0 1., K G, L M., L N. (: linear in / # Hidden ghosts appeared DC4: {Λ #, Ψ 7 } 2(S #7 ) 0 C4: Ω # 6 #7 / 7 0 ü Fixed Condition to complete Dirac procedure: det > #7 det{ω #, Ψ 7 } 0 ü We eliminated all the ghosts. ( is bounded.

27 HM, Suyama, Yamaguchi, Quadratic model with!"#, % ' Dirac matrix C1 C4 C2 C3 0 det + #, 0 det 0 0 ; All constraints are second class Healthy (2 + 4) DOFs ü 6 can be transformed to a lower-derivative 6.

28 Eliminating Ostrogradsky ghost ü!(#, #, #)!(#, #) ü!(# ), # ), # ) )!(# ), # ) ) ü!(# ) *, # ) *+,,, # ) ) HM, Suyama, ü!(#, #, #; /, /) ü!(#, #, #; / 0, / 0 ) # ü!(# ), # ), # ) ; / 0, / 0 ) # ) ü!(56, 5, 5, 5; / 0, / 0 ) HM, Noui, Suyama, Yamaguchi, Langlois, HM, Suyama, Yamaguchi, ; in prep. ü!(567, 5 7, 5 7, 5 7 ; # ), # ), # ) ; / 0, / 0 ) ü!(# 0 8 *9,, ; # 0 8:; * 0, ; ; # <, # 0 <)

29 Summary Ostrogradsky ghosts appear as! 2nd-order time derivatives $: linear in % which can be removed by degeneracy conditions. The analysis of!(' ), ' ), ' ) ; -., -. ) applies for a wide class of model buildings. We found that for quadratic model with 012, -.! 3rd-order time derivatives $: linear in %, 3 We constructed the first ghost-free model with 3rdorder time derivatives in!. The analyses of general! and field theory are work in progress.